We know that for the Hall effect of the electrons, in addition to the normal
velocity from usual band dispersion contribution, the Berry curvature induces
an anomalous velocity always transverse to the electric field, which gives
rise to a Hall current, thus the Hall effect occurs [1]}. For the magnon
Hall effect [2]} recently observed, the authors also found the
anomalous velocity due to the Berry connection which is responsible for the
thermal Hall conductivity. Therefore in this article we will derive the theory of
the PHE in a more natural way where the Berry phase effect inducing the anomalous
velocity contributes to the extra term of the heat current. Thus the Berry phase
effect is straightforward to take the responsibility of the PHE.
